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A topos topos

A quantum operation/quantum channel $chan \,\colon\, \mathscr{H} \otimes \mathscr{H}^\ast \to \mathscr{H} \otimes \mathscr{H}^\ast$ is called a noisy operation [Horodecki, Horodecki & Oppenheim 2003] or a unistochastic channel [Życzkowski & Bengtsson 2004] if it has an environmental representationhannel#QuantumChannelsAndDecoherence) where the environment/bath system $\mathscr{B}$ is in its maximally mixed state:

$$chan(\rho) \;\;=\;\; \frac{1}{dim(\mathscr{B})} \mathrm{tr}^{\mathscr{B}} \Big( U \big( \rho \otimes id_{\mathscr{B}} \big) U^\dagger \Big) \,.$$

Remark

The theorem (herechannel#QuantumChannelsAsPartialTracesOfUnitariesOnTensors)) that every endo-quantum channel has an environmental representation is often advertized with the addendum that “… and such the environment may be chose to be in a pure state”, but in fact this is assumption is what the existing proofs rely on. For environments in non-pure states it is not clear that they can environmentally represent all quantum channels, and for noisy/unistochastic channels it is not to be expected that they exhaust all quantum channels.

But Müller-Hermes & Perry 2019 show that all unital quantum channels on qbits can be realized as noisy/unistochastic channels (with a bath of size at least 4).

References

Precursor discussion of the concept is due to:

• Michal Horodecki, Pawel Horodecki, Jonathan Oppenheim, Reversible transformations from pure to mixed states, and the unique measure of information, Phys. Rev. A 67 062104 (2003) [doi:10.1103/PhysRevA.67.062104, arXiv;quant-ph/0212019]

  (who speak of “noisy operations”)

• David Poulin, Robin Blume-Kohout, Raymond Laflamme, Harold Ollivier, around (2) of: Exponential speed-up with a single bit of quantum information: Testing the quantum butterfly effect, Phys. Rev. Lett. 92 177906 (2004) [arXiv:quant-ph/0310038, doi:10.1103/PhysRevLett.92.177906]

The terminology “unistochastic channels” was introduced in:

• Karol Życzkowski, Ingemar Bengtsson, p. 13 of: On Duality between Quantum Maps and Quantum States, Open Systems & Information Dynamics 11 01 (2004) 3-42 [doi:10.1023/B:OPSY.0000024753.05661.c2]

• Ingemar Bengtsson, Karol Życzkowski, p. 259 of: Geometry of Quantum States — An Introduction to Quantum Entanglement, Cambridge University Press (2006) [doi:10.1017/CBO9780511535048, ResearchGate]

• Marcin Musz, Marek Kuś, Karol Życzkowski, Unitary quantum gates, perfect entanglers, and unistochastic maps, Phys. Rev. A 87 022111 [doi:10.1103/PhysRevA.87.022111]

Proof that all unital quantum channels on qubits are unistochastic (noisy operations) for a bath of size at least 4:

• Alexander Müller-Hermes, Christopher Perry, All unital qubit channels are 4-noisy operations, Letters in Mathematical Physics 109 (2019) 1–9 [doi:10.1007/s11005-018-1104-x, arXiv:1802.01337]

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In one direction, assume that

$$chan \,\colon\, \mathscr{H} \otimes \mathscr{H}^\ast \longrightarrow \mathscr{H} \otimes \mathscr{H}^\ast$$

is a completely positive map. Then by operator-sum decomposition there exists a set (finite, under our assumptions) inhabited by at least one element

$$s_{ini} \,\colon\, S \,,$$

and an $S$-indexed set of linear operators

$$s \,\colon\, S \;\;\; \vdash \;\;\; E_s \;\colon\; \mathscr{H} \longrightarrow \mathscr{H} \,,\;\;\;\; \text{with} \;\;\;\; \underset{s}{\sum} E_s^\dagger \cdot E_s \,=\, Id \mathrlap{\,,}$$

such that

$$chan(\rho) \;=\; \underset{s}{\sum} \, E_s \cdot \rho \cdot E_s^\dagger \,.$$

Now take

$$\mathscr{B} \,\equiv\, \underset{S}{\oplus} \mathbb{C}$$

with its canonical Hermitian inner product-structure with orthonormal linear basis $\big(\left\vert s \right\rangle\big)_{s \colon S}$ and consider the linear map

$$\array{ \mathllap{ V \;\colon\;\; } \mathscr{H} &\longrightarrow& \mathscr{H} \otimes \mathscr{B} \\ \left\vert \psi \right\rangle &\mapsto& \underset{s}{\sum} \, E_s \left\vert \psi \right\rangle \otimes \left\vert s \right\rangle \mathrlap{\,.} }$$

Observe that this is a linear isometry

$$\begin{array}{ll} \left\langle \psi \right\vert V^\dagger V \left\vert \psi \right\rangle \\ \;=\; \underset{s,s'}{\sum} \left\langle \psi \right\vert E_{s'}^\dagger E_s \left\vert \psi \right\rangle \underset{ \delta_s^{s'} }{ \underbrace{ \left\langle s' \vert s \right\rangle } } \\ \;=\; \left\langle \psi \right\vert \underset{ Id }{ \underbrace{ \underset{s}{\sum} E_{s}^\dagger E_s } } \left\vert \psi \right\rangle \\ \;=\; \left\langle \psi \vert \psi \right\rangle \mathrlap{\,.} \end{array}$$

This implies that $V$ is injective so that we have a direct sum-decomposition of its codomain into its image and its cokernel orthogonal complement, which is unitarily isomorphic to $dim(\mathscr{B})-1$ summands of $\mathscr{H}$ that we may identify as follows:

$$\mathscr{H} \otimes \mathscr{B} \;\simeq\; V\big( \mathscr{H} \big) \oplus \Big( \mathscr{H} \otimes \big( \mathscr{B} \ominus \mathbb{C}\left\vert s_0 \right\rangle \big) \Big) \,.$$

In total this yields a unitary operator

$$U \;\colon\; \mathscr{H} \otimes \mathscr{B} \,\simeq\, \mathscr{H} \oplus \Big( \mathscr{H} \otimes \big( \mathscr{B} \ominus \mathbb{C}\left\vert s_{ini} \right\rangle \big) \Big) \underoverset{}{}{\longrightarrow} V\big( \mathscr{H} \big) \oplus \Big( \mathscr{H} \otimes \big( \mathscr{B} \ominus \mathbb{C}\left\vert s_{ini} \right\rangle \big) \Big) \;\simeq\; \mathscr{H} \otimes \mathscr{B}$$

and we claim that this has the desired action if we couple the system to the pure bath state: $\left\vert s_0 \right\rangle$:

$$\begin{array}{l} trace^{\mathscr{B}} \Big( U \big( \left\vert s_{ini} \right\rangle \rho \left\langle s_{ini} \right\vert \big) U^\dagger \Big) \\ \;=\; \underset{s,s'}{\sum} trace^{\mathscr{B}} \big( \left\vert s \right\rangle E_s \cdot \rho \cdot E_{s'}^\dagger \left\langle s' \right\vert \big) \\ \;=\; \underset{s,s'}{\sum} \underset{ \delta_{s}^{s'} }{ \underbrace{ \left\langle s' \vert s \right\rangle } } E_s \cdot \rho \cdot E_{s'}^\dagger \\ \;=\; \underset{s}{\sum} E_s \cdot \rho \cdot E_{s}^\dagger \\ \;=\; chan(\rho) \,. \end{array}$$

Idea

A linear isometry is a linear map

$$\phi \,\colon\, \mathscr{H}_1 \longrightarrow \mathscr{H}_2$$

between vector spaces equipped with (Hermitian) inner products ${\langle \cdot \vert \cdot \rangle}_{i}$ (often: Hilbert spaces) which preserves these inner products, in that

$$\left\langle \phi(-) \vert \phi(-) \right\rangle_2 \;=\; \left\langle - \vert - \right\rangle_2$$

or equivalently in terms of adjoint operators:

$$\phi^\dagger \cdot \phi \,=\, Id_{\mathscr{H}_1} \,.$$

If also the reverse condition $\phi \cdot \phi^\dagger$ holds, then $\phi$ is called a unitary operator, which is the case iff $\phi$ is surjective map.

References

Environmental representation of measurement channels

By the general theorem about environmental representations of quantum channels, every quantum measurement channel on a quantum system $\mathscr{H}$ may be decomposed as

1. coupling of $\mathscr{H}$ to an environment/bath system $\mathscr{B}$,

2. unitary evolution of the composite system $\mathscr{H} \otimes \mathscr{B}$,

3. averaging the result over the environment states.

The way this works specifically for quantum measurement channels has precursor discussion von Neumann 1932 §VI.3 and received much attention in discussion of quantum decoherence following Zurek 1981 and Joos & Zeh 1985.

(independently and apparently unkowingly of the general discussion of environmental representations in Lindblad 1975)

Concretely,

(we shall restrict attention to finite-dimensional Hilbert spaces not to get distracted by technicalities that are irrelevant to the point we are after)

if $\left\vert b_{\mathrm{init}} \right\rangle \,\colon\, \mathscr{B}$ denotes the initial state of a “device” quantum system then any notion of this device measuring the given quantum system $\mathscr{H}$ (in its measurement basis $W$, $\mathscr{H} \simeq \underset{W}{\oplus}\mathbb{C}$) under their joint unitary quantum evolution should be reflected in a unitary operator under which [Zurek 1981 (1.1), Joos & Zeh 1985 (1.1.), following von Neumann 1932 §VI.3, review includes Schlosshauer 2007 (2.51)]:

1. the system $\mathscr{H}$ remains invariant if it is purely in any eigenstate $\left\vert w \right\rangle$ of the measurement basis,

2. while in this case the measuring system evolves to a corresponding “pointer state” $\left\vert b_w \right\rangle$:

(1)$$\array{ &\mathclap{ \color{green} \array{ \text{unitary} \\ \text{measurement interaction} } }& \\ \mathllap{ U_W \;\colon\;\; } \mathscr{H} \otimes \mathscr{B} &\longrightarrow& \mathscr{H} \otimes \mathscr{B} \\ \left\vert w, b_{ini} \right\rangle &\mapsto& \left\vert w, b_w \right\rangle }$$

for $b_{\mathrm{ini}}$ and $b_w$ distinct elements of an (in practice: approximately-)orthonormal basis for $\mathscr{B}$. (There is always a unitary operator with this mapping property (1), for instance the one which moreover maps $\left\vert w, b_{w}\right\rangle \mapsto \left\vert w, b_{\mathrm{ini}}\right\rangle$ and is the identity on all remaining basis elements.)

But then the composition of the corresponding unitary quantum channel with the averaging channel over $\mathscr{B}$ is indeed equal to the $W$-measurement quantum channel on $\mathscr{H}$ (cf. eg. Schlosshauer 2007 (2.117), going back to Zeh 1970 (7)), as follows: